Currently, thin-film piezoelectric membranes are known that use films of piezoelectric materials such as lead zirconate titanate (PZT) or an alloy of aluminium nitride (AlN), described in the United States patent application US 2006/0209128. The membranes may operate in two possible actuation modes: the d31 or the d33 mode.
Generally, the following equations give the coefficients d31 and d33 as a function of the strain of a material of polarization P in an electric field applied to the material:d31=S1/E3 d33=S3/E3 
with S1 corresponding to the strain along the axis 1, S3 corresponding to the strain along the axis 3 and E3 corresponding to the electric field along the axis 3 (as shown in FIG. 1).
The operating modes depend on the direction in which the material is polarized and on the direction in which an actuating electric field is applied. FIGS. 2a and 2b thus show two typical actuating structures with coefficients d31 and d33. In a d31 mode, the voltage is applied between the top electrode and the bottom electrode as shown in FIG. 2a. In the case of the d33 mode, the voltage is applied between the top electrodes, denoted + and − in FIG. 2b. 
More precisely, the structure comprises a membrane on the surface of a substrate 1, said membrane comprising a deformable piezoelectric film 3 on the surface of an elastic film 2 and furthermore actuating electrodes. In the case shown in FIG. 2a, the actuating electrodes are a bottom electrode Ebot and a top electrode Eup, between which are sandwiched the films 2 and 3. In the case of FIG. 2b the top control electrodes are denoted Eitop.
In operation, the membrane has a non-monotonic profile—that is to say that the curvature is, at a given moment, convex in certain places and concave in others. Muralt has shown (P. Muralt, M. Kohli, T. Maeder, A. Kholkin, K. Brooks, N. Setter and R. Luthier, Sensors and Actuators A, 48(2), 157 (1995)) that the profile is roughly that shown in FIG. 3, which figure shows the curvature profile of a hybrid elastic film/piezoelectric film membrane. The abscissa x=0 corresponds to the position of the centre of the membrane and the abscissa x=0.001 corresponds to the edge of the membrane in the fixed or immobilized region.
Thus, whether the membrane operates in the d33 or the d31 regime, actuating the piezoelectric material all over the membrane is counterproductive since it is essential that the piezoelectric material is made to operate identically in a curvature region of the same sign.
For a non-ferroelectric piezoelectric material unpolarized by the electric field (e.g. AlN, ZnO or quartz), the in-plane strain Sp is given by the following equation (1):
                              S          p                =                                            d              31                        ⁢            E                    =                                    -                              d                31                                      ⁢                          V                              e                p                                                                        (        1        )            
with ep being the thickness of the piezoelectric material and V the applied voltage.
It should be noted, as notably described in the paper by KyongWon Seo, JongSeon Park, HyongJung Kim et al.: Micromachined piezoelectric microspeakers fabricated with high quality AlN thin films, published in Integrated Ferroelectrics, 95 (74-82), pp. 74-82 (2007), that in the case of certain currently used crystalline materials, such as AlN, the sign of the coefficient d31 depends on the crystal orientation of the material. Thus, when the polarization direction of the AlN material is turned as shown in FIG. 2a, the coefficient d31 is negative.
It is this piezoelectric strain that causes a piezoelectric moment that deflects the membrane.
It is known to exploit a change in sign of the piezoelectric excitation, in an AlN-type material, by two separate top electrodes placed between the centre of the membrane and the edge of the membrane as shown in FIGS. 4a and 4b, which show a top view and a cross-sectional view, respectively, of a structure in which the two regions of the membrane of different curvature are excited simultaneously. Here, a d31 actuation is used in both regions but the sign of the applied voltage is reversed.
The voltages are applied between the top electrodes Eitop and the bottom electrode Ebot which remains grounded. Piezoelectric moments of different sign can therefore be applied to the convex and concave regions of the membrane, thus improving the actuation.